Boundedness of one-dimensional branching Markov processes

F. I. Karpelevich, Yu M. Suhov

Research output: Contribution to journalArticle

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Abstract

A general model of a branching Markov process on ℝ is considered. Sufficient and necessary conditions are given for the random variable M =sup max Ξk(t) t≥0 1≤k≤N(t) to be finite. Here Ξk(t) is the position of the kth particle, and N(t) is the size of the population at time t. For some classes of processes (smooth branching diffusions with Feller-type boundary points), this results in a criterion stated in terms of the linear ODE σ2(x)/2 f″(x) + a(x)f′(x) = λ(x)(1 -k(x))f(x). Here σ(x) and a(x) are the diffusion coefficient and the drift of the one-particle diffusion, respectively, and λ(x) and κ(X) the intensity of branching and the expected number of offspring at point x, respectively. Similarly, for branching jump Markov processes the conditions are expressed in terms of the relations between the integral μ(x) ∫ π(x, dy) (f(y) - f(x)) and the product λ(x)(1 -κ(x))f(x) where λ(x) and κ(x) are as before, μ(x) is the intensity of jumping at point x, and π(x, dy) is the distribution of the jump from x to y.

Original languageEnglish (US)
Pages (from-to)307-332
Number of pages26
JournalJournal of Applied Mathematics and Stochastic Analysis
Volume10
Issue number4
DOIs
StatePublished - Jan 1 1997

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics

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